Linear graphs and models - Cambridge University Press

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CHAPTER

3

Lainnedamr gordaeplhss E Whatisalineargraph?

How do we determine the slope of a straight-line graph? How do find the equation of a straight line from its graph?

L How do we sketch a straight-line graph from its equation?

How do we use straight-line graphs to model practical situations?

Drawing straight line graphs 3.1

P Plotting straight line graphs

Relations defined by equations such as

y = 1 + 2x y = 3x - 2 y = 10 - 5x y = 6x

are called linear relations because they generate straight line graphs. For example, consider the relation y = 1 + 2x. To plot a graph, we first need to form a table of values.

M x 0 1 2 3 4

y13579

We can then plot the values from the table on a set of axes, as shown opposite.

AThe points appear to lie on a straight line.

A ruler can then be used to draw in this straight

Sline to give the graph of y = 1 + 2x.

y 10

8 y = 1 + 2x

6

4

2

x

0

12345

106

Cambridge University Press ? Uncorrected Sample Pages ? 978-0-521-74049-4 2008 ? Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

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Example 1

Constructing a graph from a table of values

Plot the graph of y = 8 - 2x by forming a table of values of y using x = 0, 1, 2, 3, 4.

Solution

1 Set up table of values. When x = 0, y = 8 - 2 ? 0 = 8. When x = 1, y = 8 - 2 ? 1 = 6, and so on.

2 Draw, label and scale a set of axes to cover all values.

LE 3 Plot the values in the table on the graph by marking with a dot (?). The first point is (0, 8). The second point is (1, 6), and so on.

P 4 The points appear to lie on a straight line.

Use a ruler to draw in the straight line.

AM Label the line y = 8-2x.

x0 1 234 y86420

y

10 8 6 4 2

x O 1 2 345

y

10 8 6 4 2

x O 1 2 345

y

10

8

6

y = 8 ? 2x

4

2

x O 1 2 345

A graphics calculator can also be used to draw straight-line graphs, although it can take some

Sfiddling around with scaling to get just the graph you want. One bonus of using a graphics

calculator is that, in drawing the graph, it automatically generates a table of values for you.

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How to draw a straight-line graph and show a table of values using the TI-Nspire CAS

Use a graphics calculator to draw the graph y = 8 ? 2x and show a table of values. Steps

1 Start a new document (by pressing / + N)

and select 2: Add Graphs & Geometry.

2 Type in the equation as shown. Note that f 1(x) represents the y term. Press enter to obtain the graph below.

Hint: / + will hide the entry line to give more

E screen view. L 3 Change the window setting to see the P key features of the graph. Press

b/4:Window/1:Window Settings and edit as

shown. Use the key to move between the entry lines. Press enter when you have finished editing the settings. The re-scaled graph is shown below.

M 4 To show values on the graph, press: b/5:Trace/1:Graph Trace and then use arrows to move along the graph.

A5 To show a table of values, press / + N. SUse the arrows to scroll through the values

in the table.

Cambridge University Press ? Uncorrected Sample Pages ? 978-0-521-74049-4 2008 ? Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

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How to draw a straight-line graph and show a table of values using the ClassPad

Use a graphics calculator to draw the graph of y = 8 - 2x and show a table of values. Steps 1 Open the built-in Graphs

and Tables application. 2 Tap the 6 icon and

complete the View Window as shown to get a graph more like the one plotted by

E hand.

Notes: 1 Making x min and y min = -0.5,

rather than zero, enables you to see the axes.

L 2 The dot value gives the trace increment for the graph and is set automatically.

3 Enter the equation into the graph editor window by

P typing 8 - 2x and then

pressing E. Tap the $ icon to plot the graph. 4 Tapping resize (r) from the toolbar increases the

M size of the graph window.

Selecting Analysis from the menu and then Trace will place a cursor on the graph

Aand the equation will be

displayed in a window at the bottom of the screen. The coordinates of the point

S(2.25, 3.5) are also shown.

Cambridge University Press ? Uncorrected Sample Pages ? 978-0-521-74049-4 2008 ? Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

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5 Tapping the icon from

the toolbar will display a

table of values.

Tapping the 8 icon from

the toolbar opens the Table

Input dialog box. The values displayed in the table can be adjusted by changing the values in this window.

Exercise 3A E 1 Two straight-line graphs, y = 4 + x and

y

L y = 8 - 2x, are plotted as shown opposite.

10

a Reading from the graph of y = 4 + x,

determine the missing coordinates:

8

(0, ?), (2, ?), (?, 7), (?, 9).

P b Reading from the graph of y = 8 - 2x,

6

determine the missing coordinates:

4

(0, ?), (1, ?), (?, 4), (?, 2).

2

y = 4 + x y = 8 ? 2x

x

0

12345

M 2 Plot the graph of the linear equations below by first forming a table of values of y using x = 0, 1, 2, 3, 4.

a y = 1 + 2x

b y =2+x

c y = 10 - x

d y = 9 - 2x

3 For each of these linear equations, use a graphics calculator to do the following:

Ai Plot a graph for the window given.

ii Generate a table of values.

a y =4+x

S-10 x 10

b y = 2 + 3x -0.5 x 5

c y = 10 + 5x -0.5 x 5

-10 y 10

-0.5 y 20

-0.5 y 40

d y = 5x

e y = -5x

f y = 100 - 5x

-5 x 5

-5 x 5

-0.5 x 25

-25 y 25

-25 y 25

-25 y 125

Cambridge University Press ? Uncorrected Sample Pages ? 978-0-521-74049-4 2008 ? Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

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3.2 Determining the slope of a straight line

Positive and negative slopes

One of the things that make one straight-line graph

y

look different from another is its steepness or slope. Another name for slope is gradient.

10 A

For example, the three straight lines on the graph opposite all cut the y-axis at y = 2, but they have quite different slopes.

Line A has the steepest slope while Line C has the gentlest slope. Line B has a slope somewhere in between.

In all cases, the lines have positive slopes; that

E is, they rise from left to right. Similarly, the three straight lines on the graph opposite all cut the y-axis at y = 10, but they have quite different slopes.

L In this case, Line D has the gentlest slope

while Line F has the steepest slope. Line E has a slope somewhere in between.

In all cases, the lines have negative slopes;

P that is, they fall from left to right.

8

B

6 C

4

2 ?1 0

x 12 345

y

10 D

8

6

4 2 ?1 0

E

F

x

12 345

Giving slope a value: the slope

When talking about the slope of a line (Line C, for example), we want to be able to do more

M than say that it has a gentle positive slope. We would like to be able to give the slope a number

that reflects this fact. We do this as follows.

First, two points A and B on the line are chosen.

As we go from A to B (left to right), we move:

B

a distance vertically, called the rise

Aa distance horizontally, called the run.

The slope of the line is found by dividing the

Srise by the run.

Rise A

Run

Slope

=

rise run

Example 2

Finding the slope of a line from a graph: positive slope

Find the slope of the line through the points (1, 4) and (4, 8).

Cambridge University Press ? Uncorrected Sample Pages ? 978-0-521-74049-4 2008 ? Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

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Solution

y

Slope = rise run

10

Rise = 8 - 4 = 4

(4, 8) 8

Run = 4 - 1 = 3

Slope = 4 = 1.33 (correct to 2 d.p.) 3

6 (1, 4)

4

2

Run = 3

0

12 3

Rise = 4

x 45

E Example 3

Finding the slope of a line from a graph: negative slope

Find the slope of the line through the points (0, 10) and (4, 2).

Solution

L Slope = rise run Rise = 2 - 10 = -8 Run = 4 - 0 = 4

P Slope = -8 = -2 4

y

(0, 10) 10

Run = 4 8

6

4

2

Rise = ?8 (4, 2)

x

Note: In this example, we have a negative `rise' or a `fall'.

0

12 345

M A formula for finding the slope of a line

While the `rise/run' method for finding the slope of a line will always work, some people

prefer to use a formula for calculating the slope. The formula is derived as follows.

ALabel the coordinates of point A: (x1, y1).

Label the coordinates of point B : (x2, y2).

B(x2, y2)

Slope = rise run

SRise = y2 - y1

A(x1, y1)

Rise = ( y2 ? y1)

Run = x2 - x1

Run = x2 ? x1

Slope

=

y2 x2

- -

y1 x1

Slope = y2 ? y1 x2 ? x1

Cambridge University Press ? Uncorrected Sample Pages ? 978-0-521-74049-4 2008 ? Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

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Example 4

Finding the slope of a line using the formula

Find the slope of the line through the points (1, 7) and (4, 2) using the formula.

Solution

y

Slope

=

y2 x2

- -

y1 x1

Let (x1, y 1) = (1, 7) and (x2, y 2) = (4, 2).

Slope

=

2-7 4-1

=

-1.67

(to

2

d.p.)

E Summary L A straight-line graph that rises from

left to right is said to have a positive slope (positive rise).

y

P Positive slope

x O

M A straight-line graph that is horizontal

has zero slope (`rise' = 0).

y

AZero slope x

SO

10

8

(1, 7)

6

4

(4, 2) 2

x 0 12345

A straight-line graph that falls from left to right is said to have a negative slope (negative rise).

y

Negative slope

x O

The slope is not defined for a straight-line graph that is vertical.

y

Slope not defined

x O

Cambridge University Press ? Uncorrected Sample Pages ? 978-0-521-74049-4 2008 ? Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

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